Homeostasis at different backgrounds: The roles of overlayed feedback structures in vertebrate photoadaptation

We have studied the resetting behavior of eight basic integral controller motifs with respect to different but constant backgrounds. We found that the controllers split symmetrically into two classes: one class, based on derepression of the compensatory flux, leads to more rapid resetting kinetics as backgrounds increase. The other class, which directly activates the compensatory flux, shows a slowing down in the resetting at increased backgrounds. We found a striking analogy between the resetting kinetics of vertebrate photoreceptors and controllers based on derepression, i.e. vertebrate rod or cone cells show decreased sensitivities and accelerated response kinetics as background illuminations increase. The central molecular model of vertebrate photoadaptation consists of an overlay of three negative feedback loops with cytosolic calcium (Cai2+), cyclic guanosine monophosphate (cGMP) and cyclic nucleotide-gated (CNG) channels as components. While in one of the feedback loops the extrusion of Cai2+ by potassium-dependent sodium-calcium exchangers (NCKX) can lead to integral control with cGMP as the controlled variable, the expected robust perfect adaptation of cGMP is lost, because of the two other feedback loops. They avoid that Cai2+ levels become too high and toxic. Looking at psychophysical laws, we found that in all of the above mentioned basic controllers Weber’s law is followed when a “just noticeable difference” (threshold) of 1% of the controlled variable’s set-point was considered. Applying comparable threshold pulses or steps to the photoadaptation model we find, in agreement with experimental results, that Weber’s law is followed for relatively high backgrounds, while Stephens’ power law gives a better description when backgrounds are low. Limitations of our photoadaption model, in particular with respect to potassium/sodium homeostasis, are discussed. Finally, we discuss possible implication of background perturbations in biological controllers when compensatory fluxes are based on activation.

In the second part of the paper the authors focus on the photoreceptor system in vertebrates. This system contains 3 overlayed feedback loops whose the molecular components are well described. They involve calcium and cGMP, as well as the CNG channel and NCKX pump. After an estimation of the parameter values based on experimental observations, the authors study the adaptation of the system to pulse and to step perturbations. For the pulse perturbation, the authors found that the threshold (light stimulus needed to get a response of a given amplitude) -background relationship follows a Weber law for large background and a Stephen power law for small background, in agreement with the experimental observations. For the step perturbation, they found that the amplitude of the cGMP excursion decreases monotonically with the increase of the background (as expected) but that the response time first decreases and then increases with the background. This latter, non-intuitive prediction is related to observations done on turtle photoreceptors. Finally, the authors analyse in detail the role of the feedback loops. They found that 2 antagonistic feedbacks allow a robust adaptation while a third, negative feedback impairs the robustness of the adaptation but contributes to keep calcium level low (thus preventing high cytotoxic levels).
The paper is well structured and clearly written. The authors provide an elegant way to understand the kinetics of the vertebrate photoreceptor system and made clear links with experimental observations. The analysis of the various motifs provide a solid basis to understand the "logic" of the photoreceptor system but possibly also other systems. The codes to reproduce each figure are made available.
Response: We would like to thank the reviewer for the positive assessment.
Reviewer #1: "I was just wondering if other models exist for the photoreceptor system in vertebrates and how they di↵er from the present one, regarding the molecular components taken into account and/or the kinetics." Response: Yes, there is an extensive literature on modeling retinal photoadaptation. We agree that a comparison of our 'homeostatic model' with previous calculations is in place. We have therefore added near the end (just before the 'Conclusion and outlook' section) the section 'Other model approaches'. It gives an overview of previous model calculations. Since none of the earlier calculations/models used a 'homeostatic approach', we added, at the end of this section the following paragraph: "Our approach, although primarily kinetic in nature, di↵ers from previous adaptation models by looking at photoadaptation from a robust homeostatic viewpoint. In this respect we agree with Billman [57] that homeostatic approaches are still underappreciated and are far too often ignored as a central organizing principle in physiology."

Changes made in the revised manuscript based on the comments by Reviewer #2
Changes made in the manuscript with respect to the comments by Reviewer #2 are indicated in blue in the marked-up copy of the revised manuscript.
Reviewer #2: "In the article entitled "Homeostasis at di↵erent backgrounds: the roles of overlayed feedback structures in vertebrate photoadaptation" by Grini et al., the authors classify the resetting behavior of eight possible basic integral regulatory motifs into two classes and discuss how the multiple feedback loops in molecular models of vertebrate photoadaptation are involved in controlling homeostasis. The work is original, has not been reported elsewhere, and the methods and analysis are of a high standard. The description is also adequate. However, some parts of the paper are confusing and there are several points that should be revised." Response: Thank you for your positive assessment and for your comments on how the manuscript can be improved. We have revised the manuscript at several places according to your comments, and describe the made changes in more detail below.
Reviewer #2: "First, this reviewer's main concern is that this article should be divided into two separate studies, the study on the classification of basic integral controller motifs, and vertebrate photoadaptation molecular model dynamics. It is one outcome to discuss the existence of multiple feedback loops in the molecular model of vertebrate photoadaptation, each of which shows similarities to integral controller motifs, and their molecular dynamics. On the other hand, the finding that the possible basic integral controller motifs can be divided into two classes is also an achievement. Each of these is an independent achievement, but combining them into a single paper has blurred the point of contention. While it is possible to publish as is by addressing the following concerns, this reviewer believes that one option is to separate the independent results and publish them as two concise papers." Response: Although splitting this manuscript into two separate parts may be an option, we do not wish to do so. The findings on the controller motifs are not independent of our proposed 3-feedback light adaptation model. They provide the actual basis for it. We further feel that a splitting into two separate manuscripts will lead to unnecessary fragmentation of our work, which we want to avoid. However, as a response to the Academic Editor's comment, we have now explicitly described that the paper comes in two parts: one related to the analysis of the basic controller motifs, and another one, where retinal photoreceptor responses are analyzed on basis of the controller motifs responses, which leads to the 3-feedback light adaptation model. Please, see our changes made in response to the comment by the Academic Editor.
Reviewer #2: "The reviewer was di cult to understand the meaning and significance of psychophysical laws (Weber's law and Stevens' power law). For example, there was a statement "Weber's law is followed when a "just noticeable di↵erence" (threshold) of 1% of the controlled variable's set-point was considered." in the main text. What is the significance of following Weber's law? Similarly, the reviewer could not understand the significance of being able to express this in terms of Stevens' power law. These points could be explained in more detail for the unfamiliar reader." Response: We feel it is of interest that the eight controller motifs follow Weber's and Stephens' law, since these laws are also found in experimental adaptation studies (see f eks. part IV in the textbook by Kandel et al,Ref. 21). In order to make the psychophysical laws more accessible to the unfamiliar reader, we have included in the section 'Psychophysical laws' a brief introduction. In addition, references are included such that the reader can get a more concise view.
Minor comments by Reviewer #2: • "p.2, Line 47: The constant C is not present in Eq 2." Response: C is now included.
• "p.2, Line 47: Isn't w the reference wight? Why the sudden change to general stimulus here." Response: yes, in the example w was weight, but can in fact be any other stimulus. We have now included the sentence: "Instead of weight, w can generally be any other stimulus." • "p.7, Line 170: The description of (Fig 1) after m2 and m8 is desirable." Response: We have now included ' (Fig 1)' after m2 and m8.
• "p.8, Line 193: Major points have been made, but at least a brief mention should be made of Figs 8c and 8d." Response: Yes, we agree. We have now added the sentence: "Figs 8c and d show how A max and t max depend on the background k 4 , respectively." • "p.8, Line 194: The reviewer believes this subsection should be moved to supporting information S3 Text, including Fig 9 because it is a little o↵ the main subject." Response: Antithetic control is an alternative kinetic mechanism to ensure integral control within a negative feedback loop to ensure robust homeostasis. The concept of antithetic control was developed by M. Khammash's group in Basel. Since we use antithetic control later in the photoadaptation model (Fig 13) to describe the transport of Ca 2+ and K + out of the cell by potassium-dependent sodium-calcium exchangers (NCKX), we wish to emphasize this control mode explicitly and therefore have the scheme in Fig 9 in the main manuscript. There is also the point we make that antithetic and zero-order controllers behave dynamically identical as outlined in the main paper, but shown in detail in S3 Text. To motivate having the antithetic control scheme in the manuscript, we have changed the first sentence in section 'Controller m2 with antithetic integral control' to: "Since we later will use bimolecular (antithetic) control [14,19] to describe the simultaneous removal of Ca 2+ and K + out of a photoreceptor cell by potassium-dependent sodium-calcium exchangers (NCKX), we illustrate here how scheme m2 works with antithetic integral control (Fig 9)." • "p.9, Line 205: Eq 12, not Eq 11?" Response: Please note that Eq 12 is Eq 11, but now under zero-order conditions. This defines A set , as outlined in Eq. 12. The reference to Eq 12 on line 205 (line 225 in marked-up copy) is correct.
• "p.9, Line 222: This part should also be briefly explained at least for Fig 11 c and 11d." Response: We have now added the sentence: "Figs 11c and d show how A max and t max depend on the background k 3 , respectively." • "p.11, Line 247: CNG channels are not familiar to the general reader. Appropriate references should be cited." Response: With respect to CNG channels we have now cited the textbook by Marks et al. and the research paper by Hsu and Molday. The latter work is later used to extract rate parameters for the photoadaptation model.
• "p.12, Eq 21: This notation should correspond to the notation with vleak in Fig 13." Response: We have now written v leak in Eq 21 as in Fig 13, and at two other places.
• "p.13, Line 310: Eq 11 wrong?" Response: Reference to Eq 12 is correct, since this is the zero-order version (with respect to E) of Eq 11.
• "p.13, Lines 307-312: It is interesting . . . possibly toxic Ca2+ concentration. I feel that these sentences are o↵ the main line. Is it really necessary?" Response: We feel this is indeed an interesting aspect to discuss. As robust perfect adaptation could be considered to be an evolutionary goal by itself, we have here the situation where the Ca inhibition of the CNG channel 'destroys' the robust perfect adaptation of cGMP (induced by the two other Ca-related feedback loops). The role of Ca inhibition of the CNG channel is apparently to avoid too high cytosolic Ca levels. It is well documented that too high Ca levels lead to apoptosis, i.e. to cell death. To substantiate this aspect, we have slightly rewritten the last sentence of this section and cited the work by He et al. The experimental data we used were presented by their respective authors as normalized data, either in % or on the scale from 0-1. In order to make this fact more clear, we added the word 'Normalized' in the beginning of the legend of Fig 14. • "p.13, Line 314: Where did "experimental values" come from? Please cite the exact reference." Response: We have now cited work (Refs 32,34,47,48), where cGMP and Ca 2+ i concentrations have been estimated.
• "p.16, Lines 378-379: "Instead of . . . Ca2+ and K+ [16]." Does this sentence make sense? Shouldn't the extra information in the middle be removed because it makes it di cult to understand the main points?" Response: The original papers on antithetic control (Ref 14, 15) considered uncatalyzed systems. We have recently considered catalyzed antithetic control (Ref 16). Eqs 21 and 22 write the antithetic control of cGMP by Ca 2+ in an uncatalyzed transport of Ca 2+ and K + out of the cell. Since the NCKX transporter actually catalyzes the transport, we simply wanted to say that one could have also used an catalyzed antithetic controller as described in Ref 16 on general grounds. We agree that the sentence was di cult to